Answer:
Given side BC is congruent with side DC and side AC is congruent with EC, we can prove that triangle BCA is congruent with triangle DCE using the following steps:
Step 1: By using the definition of congruent, we know that if two sides of a triangle are congruent, then the angles opposite to those sides are also congruent. Therefore, angle BCA is congruent to angle DCE, and angle BAC is congruent to angle DEC.
Step 2: Since we have two pairs of congruent angles, we can use the angle-angle similarity postulate to prove that triangle BCA is similar to triangle DCE.
Step 3: Since both of the triangles are similar, we can use the third congruent side AC which is congruent to EC, to prove that the two triangles are congruent.
Step 4: Now, we can use the side-side-side postulate, which states that if three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent.
Therefore, we can conclude that triangle BCA is congruent to triangle DCE.
Proof:
By the definition of congruence, we know:
-BC is congruent to DC
-AC is congruent to EC
By the angle-angle similarity postulate, we know that:
-angle BCA is congruent to angle DCE
-angle BAC is congruent to angle DEC
By the side-side-side postulate, we know that:
-triangle BCA is congruent to triangle DCE
Therefore, we can prove that triangle BCA is congruent to triangle DCE.